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Full-Text Articles in Algebra
Experimenting With The Identity (Xy)Z = Y(Zx), Irvin Roy Hentzel, David P. Jacobs, Sekhar V. Muddana
Experimenting With The Identity (Xy)Z = Y(Zx), Irvin Roy Hentzel, David P. Jacobs, Sekhar V. Muddana
Irvin Roy Hentzel
An experiment with the nonassociative algebra program Albert led to the discovery of the following surprising theorem. Let G be a groupoid satisfying the identity (xy)z = y(zx). Then for products in G involving at least five elements, all factors commute and associate. A corollary is that any semiprime ring satisfying this identity must be commutative and associative, generalizing a known result of Chen.
Rings With (A, B, C) = (A, C, B) And (A, [B, C]D) = 0: A Case Study Using Albert, Irvin R. Hentzel, D. P. Jacobs, Erwin Kleinfeld
Rings With (A, B, C) = (A, C, B) And (A, [B, C]D) = 0: A Case Study Using Albert, Irvin R. Hentzel, D. P. Jacobs, Erwin Kleinfeld
Irvin Roy Hentzel
Albert is an interactive computer system for building nonassociative algebras [2]. In this paper, we suggest certain techniques for using Albert that allow one to posit and test hypotheses effectively. This process provides a fast way to achieve new results, and interacts nicely with traditional methods. We demonstrate the methodology by proving that any semiprime ring, having characteristic ≠ 2, 3, and satisfying the identities (a, b, c) - (a, c, b) = (a, [b, c], d) = 0, is associative. This generalizes a recent result by Y. Paul [7].